I have an equation in complex domain,
$$P(e^u,e^v)=\sum_{i=1}^{N} a_i e^{b_i u + c_i v}=0 \;\;\;\text{(A)}$$
and by redefining, at the roots (I'm only showing work for one root), the first coordinate $$u \mapsto u_{\text{coord. at root}} + z^2,$$ where $u_{\text{coord. at root}}$ is just a number, I then have
$$\tilde{P}(z,e^v)=\sum_{i=1}^{N} a_i e^{b_i (z^2+d_i) + c_i v}=0 \;\;\;\text{(B)},$$
where $d_i$ is just $u_{\text{coord. at root}}$, and now I wish to find an expression for $v(z)$. The numbers $a_i$, $b_i$, $c_i$, and $d_i$ are just complex numbers. I also know the non degenerate roots of this equation, which manifest as $z \to 0$ in the following (only one root shown)(when $z=0$, this 2-tuple satisfies $P(e^u,e^v)=0$ identically);
$$\left[ u={{5\,\log 3}\over{2}} + z^2, v=\log \left(-\sqrt{3}\right)- \log 3 \right] $$
I thought about trying to apply Lagrange's Reversion theorem (Wikipedia or Goursat [see Wikipedia reference https://en.wikipedia.org/wiki/Lagrange_reversion_theorem]) as a way to accomplish this. This seems most likely yet I fail to see how to get it into the appropriate form (assuming all in the proof of this theorem). On this page, it has $$v = x + y\,f(v),$$ with $$v=v(x,y).$$ Then one has the series representation, and for $g(v)=v$ $$v = x + \sum_{k=1}^{\infty} \frac{y^k}{k!} \left( \frac{\partial}{\partial x}\right)^{k-1} \left( (f(x))^k \right) .$$ Here, as $$y \to 0,$$ $$v \to x.$$
In my case I could say that as $$z \to 0,$$ $$v \to {\text{v-coord of root (known)}}.$$ So my idea of applying this here is to say I have the result $$ v = {\text{v-coord of root corresponding to u-coord, $d_i$}} + z\,P(z,e^v) .$$ But in all the examples I see, $f(v)$ is in terms of one variable, and $y$ is not in $f$ explicitly, e.g. Kepler's equation, $$M = E-e\,\sin(E),$$ where $e$ and $E$ are separable. So I'm not sure $P(z,e^v)$ should be the last piece of the model equation, nor am I sure I can even use this Lagrang'e Reversion theorem.
I also saw Lagrange's Inversion theorem but I can't isolate either variable...
I was also thinking of the iterative method for an asymptotic expansion (like in chapter one of Hinch's book, Perturbation Methods) but, again, I can't really isolate one of the variables, let alone the one I want ($v$).
I read in John Boyd's book, Solving Transcendental Equations, about how one could express a function as a polynomial, and so I thought maybe converting all my exponentials into trigonometric functions. Then I could possibly express the function as a polynomial in Chebyshev basis polynomials and then use some algorithm to find a root (maybe Clenshaw?). But I really don't see how this is going to get me where I want to be.
Asymptotics, perturbation,...
Does anyone have any suggestions?